So there are 12 pentagons and 20 patches of a triangular cutout of a hexongal mesh mapped to the sphere. When five equilateral triangles meet in one corner and cover the whole sphere, this can only be the configuration induced by a icosahedron. You only need to implement truncate and dual operators for the shape you are looking for.įirst some analysis of the image in the question: the spherical triangle spanned by neighbouring pentagon centers seems to be equilateral. such as winged-edge or half-edge datastructures for your mesh. I would suggest using a datastructure that makes it easy to traverse the neighbourhood give a vertex, edge etc. You can already see where this is going.Īpply steps 3 & 4 repeatedly until you are satisfied.įor example below is the mesh for dtdtdtdtI. At this point the recipe is tdtI (read from right!). We apply the "Dual" operator (Conway notation d). We apply a "Truncate" operation (Conway notation t) to the mesh (the sperical mapping of this one is a football). The polyhedron you are looking for can be generated from an icosahedron - Initialise a mesh with an icosahedron.
The construction is easy to follow step by step, you can click the images below to get a live preview. The ( rather elegant) algorithm to generate this (and many many more) can be succinctly encoded in something called a Conway Polyhedron Notation. The shape you have is one of so called "Goldberg polyhedra", is also a geodesic polyhedra.
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